A lattice-theoretic characterization of pure subgroups of abelian groups (Q6054813)

Let \(G\) be an abelian group, a subgroup \(H \leq G\) is pure if \(G^{n}\cap H=H^{n}\) for all \(n \in \mathbb{N}\). The aim of the paper under review is to define a sublattice embedding property which, into the universe of subgroup lattices of abelian groups, is satisfied precisely by all the elements corresponding to pure subgroups. This paper is motivated by that of \textit{S. N. Chernikov} [Am. Math. Soc., Transl., II. Ser. 17, 117-152 (1961; Zbl 0131.25303)] where he describes the structure of abelian groups whose pure subgroups admit a complement. An immediate consequence of the result of this note is that any projectivity between abelian groups preserves the property considered by Chernikov.